On a σ-ideal of compact sets

Tukey reduction among analytic directed orders

“…In point (i), N N < T E µ and E µ ≤ T NWD were proved by Fremlin [4] with strictness of the latter inequality following from Theorem 4.2(ii) and (iii) below and established in [4], [13], and [20]. The inequality NWD < T I 0 in point (i) was proved independently by Mátrai [12] and by Moore and Solecki [14]. In point (ii), N N < T Z 0 is due to Isbell [7], Z 0 ≤ T ℓ 1 to Fremlin [4], and ℓ 1 ≤ T Z 0 to Louveau and Veličković [9].…”

“…Point (iii) was proved independently by Mátrai [13] and Solecki-Todorcevic [20]. Point (iv) was proved by Mátrai in [12]. Strictly speaking, it was not proved there for I 0 , but for the σ-ideal defined in [11].…”

“…The first theorem shows the Tukey comparisons between elements of the same class: P-ideals and σ-ideals. (i) ( [4], [12], [13], [14], [20])…”

Survey on the Tukey theory of ultrafilters

“…It was employed by Day in [7], Isbell in [15] and [16], and Todorčević in [31] and [32] to find rough classifications of partially ordered sets, a setting where isomorphism is too fine a notion to give any reasonable classification. More recently, Tukey reducibility has proved to be foundational in the study of analytic partial orderings (see [29], [30] and [21]). This background will be discussed in some more detail below.…”

Avoiding families and Tukey functions on the nowhere-dense ideal

“…Two directed orders are Tukey equivalent precisely when they are isomorphic to cofinal subsets of the same directed order. For more background on Tukey reductions, we direct the reader to [3] and [14] and for applications of this notion in the study of partial orders, in addition to these two papers, the reader may consult [4], [5], [7], [8], [9], [13], [15] and [17].…”